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Looking for stabilizers in NSOP$\_1$

Yvon Bossut

TL;DR

The paper develops a Kim-stabilizer approach for groups definable in $NSOP_1$ theories, addressing how generic-type analogues behave in this context and how stabilizers interact with Kim-forking. It constructs a framework combining strengthened independence theorems, $S1$-ideals, and definable measures to show that a type-definable subgroup of a group $G$ of bounded index embeds finitely-to-one into an algebraic group over a field $F$ under suitable assumptions. The results are then extended to definable groups in $ ext{$oldsymbol{ omeant}$}$-free PAC fields of characteristic $0$, where the hypotheses hold and the embedding into an algebraic group can be realized, yielding definable amenability and isogeny-type connections. The work also clarifies limitations of the Kim-stabilizer via counterexamples and raises several questions about genericity, stability, and measure expansion in the NSOP_1 setting, highlighting both the potential and the boundaries of stabilizer-based approaches in this broader model-theoretic landscape.

Abstract

In this work we study some examples of groups definable and type-definable in NSOP1 theories. We exhibit some behaviors of these groups that differ from the ones of simple groups. We take interest in the notions of generics and stabilizers, and define the Kim-stabilizer. We apply the notion of Kim-stabilizer and the stabilizer from Hrushovsky to the context of a group G definable in an NSOP1 field F satisfying some assumptions to show that there is a finite to one embedding of a type definable subgroup of G of bounded index into an algebraic group over F. We then show that definable groups in omega-free PAC fields satisfy these conditions.

Looking for stabilizers in NSOP$\_1$

TL;DR

The paper develops a Kim-stabilizer approach for groups definable in theories, addressing how generic-type analogues behave in this context and how stabilizers interact with Kim-forking. It constructs a framework combining strengthened independence theorems, -ideals, and definable measures to show that a type-definable subgroup of a group of bounded index embeds finitely-to-one into an algebraic group over a field under suitable assumptions. The results are then extended to definable groups in oldsymbol{ omeant}-free PAC fields of characteristic , where the hypotheses hold and the embedding into an algebraic group can be realized, yielding definable amenability and isogeny-type connections. The work also clarifies limitations of the Kim-stabilizer via counterexamples and raises several questions about genericity, stability, and measure expansion in the NSOP_1 setting, highlighting both the potential and the boundaries of stabilizer-based approaches in this broader model-theoretic landscape.

Abstract

In this work we study some examples of groups definable and type-definable in NSOP1 theories. We exhibit some behaviors of these groups that differ from the ones of simple groups. We take interest in the notions of generics and stabilizers, and define the Kim-stabilizer. We apply the notion of Kim-stabilizer and the stabilizer from Hrushovsky to the context of a group G definable in an NSOP1 field F satisfying some assumptions to show that there is a finite to one embedding of a type definable subgroup of G of bounded index into an algebraic group over F. We then show that definable groups in omega-free PAC fields satisfy these conditions.

Paper Structure

This paper contains 19 sections, 38 theorems, 3 equations, 4 figures.

Table of Contents

  1. Introduction
  2. NSOP$_1$ groups
  3. Kim-generics
  4. The Kim-Stabilizer
  5. Strengthenings of the Independence Theorem in NSOP$_1$
  6. Defining the Kim-stabilizer
  7. Examples
  8. A Counter-Example
  9. Stabilizer and expansion of the language
  10. S1 ideals of formulas
  11. The assumptions on $F$
  12. Definable measures and (H3)
  13. The group configuration
  14. Germs and the Hrushovski-Weil theorem
  15. Obtaining an algebraic group
  16. ...and 4 more sections

Key Result

Theorem 1.0.0.1

hrushovski1994groups Let $G$ be a group definable in a pseudo-finite field $F$. Then $G$ is virtually definably isogenous to the set of $F$-rational points of an algebraic group over $F$.

Figures (4)

  • Figure 1: The $\mathrel{ \mathop{ \vcenter{ \hbox{\oalign{{}$\vert$\cr {} $\smile$\cr{}}} } }\displaylimits_{} } ^{\Gamma}$-generic
  • Figure 2: The group configuration
  • Figure 3: A copy of the configuration over the line $(b'z'y')$
  • Figure 4: $K \mathrel{ \mathop{ \vcenter{ \hbox{\oalign{{}$\vert$\cr {} $\smile$\cr{}}} } }\displaylimits_{} } ^{ACF}_{k}k'$, $K/k$ and $k'/k$ are regular

Theorems & Definitions (90)

  • Theorem 1.0.0.1
  • Remark 2.1.0.1
  • Definition 2.1.0.2
  • Remark 2.1.0.3
  • Proposition 2.1.0.4
  • proof
  • Lemma 2.1.0.5
  • proof
  • Remark 2.1.0.6
  • Theorem 2.2.1.1
  • ...and 80 more